Band excitation method applicable to scanning probe microscopy

ABSTRACT

Methods and apparatus are described for scanning probe microscopy. A method includes generating a band excitation (BE) signal having finite and predefined amplitude and phase spectrum in at least a first predefined frequency band; exciting a probe using the band excitation signal; obtaining data by measuring a response of the probe in at least a second predefined frequency band; and extracting at least one relevant dynamic parameter of the response of the probe in a predefined range including analyzing the obtained data. The BE signal can be synthesized prior to imaging (static band excitation), or adjusted at each pixel or spectroscopy step to accommodate changes in sample properties (adaptive band excitation). An apparatus includes a band excitation signal generator; a probe coupled to the band excitation signal generator; a detector coupled to the probe; and a relevant dynamic parameter extractor component coupled to the detector, the relevant dynamic parameter extractor including a processor that performs a mathematical transform selected from the group consisting of an integral transform and a discrete transform.

STATEMENT AS TO RIGHTS TO INVENTIONS MADE UNDER FEDERALLY-SPONSOREDRESEARCH OR DEVELOPMENT

This invention was made with United States Government support underprime contract No. DE-AC05-00OR22725 to UT-Battelle, L.L.C. awarded bythe Department of Energy. The Government has certain rights in thisinvention.

BACKGROUND INFORMATION

1. Field of the Invention

Embodiments of the invention relate generally to the field ofmeasurement apparatus and methods. More particularly, embodiments of theinvention relate to the apparatus and methods of scanning probemicroscopy.

2. Discussion of the Related Art

Prior art scanning probe microscopy apparatus and methods are known tothose skilled in the art. For instance, a conventional scanning probemicroscope is shown in FIG. 1. A sample 100 is placed on a mount and acantilever sensor 101 with a sharp tip is brought close to the samplesurface. Interactions between the sample surface and the sensor tip leadto flexural and torsional deflections of the cantilever, the magnitudeof which can vary from sub-nanometer to hundred nanometer rangedepending on operation mode. A laser 102 is deflected off the top of thecantilever into a detector 103, which is connected to driving circuitry104. Other realizations of deflection sensors based on piezoresistive,piezoelectric, capacitive, MOSFET, tuning fork, double tuning fork, andother position sensors are well known. Current flowing through thecantilever or cantilever-surface capacitance can be other modes ofdetection. Typically, the sample surface is scanned point by point toobtain the topography or functional properties of the surface.Alternatively, the response is measured in a single point as a functionof probe-surface separation, probe bias, etc, constituting spectroscopicmodes of operation.

A problem with this technology has been that methods existing to dateare generally based on the detection of the signal under a constantexcitation or at a periodic excitation at a single frequency. In theconstant or static mode, static tip deflection (or other staticparameter such as tip-surface dc current) is used to serve as a feedbacksignal to maintain constant tip-surface separation or propertymeasurement. In the periodic excitation mode, the amplitude or phase ofcantilever oscillations or other oscillatory response is selected usinglock-in amplifier or similar circuit and used as a feedback or detectionsignal. In the frequency tracking modes, the cantilever or other sensoris kept at a corresponding mechanical resonance, and changes in thedynamic characteristics of oscillation (e.g. resonant frequency oramplitude at the resonance) are detected and used as feedback ordetected signals. All these modes severely limit the amount ofinformation obtainable by the scanning probe microscope. Therefore, whatis required is a solution that provides maximum information about thetip-surface interactions.

A number of SPM techniques (e.g. Force Volume Imaging, Pulsed Force Modeand Molecular Recognition Mode) are based on the specially designedlarge-amplitude waveforms that probe different parts of theforce-distance curve to distinguish short-and long range interactions.These methods also have similar limitations, since either static force(force-distance measurements per se) or response at single frequency ismeasured at different positions of the probe tip with respect to thesurface.

The fundamental problem with this technology has been that the resonancefrequency, amplitude and quality factor (Q-factor) of the cantilevervibrating in contact with the surface under constant mechanicalexcitation, the three parameters that provide the complete descriptionof the system in the simple-harmonic oscillator approximation, cannot beunambiguously separated. Therefore, what is also required is a solutionthat allows separation of these parameters.

One unsatisfactory approach, in an attempt to solve the above-discussedproblems involves sweeping the excitation frequency at each samplepoint. However, a disadvantage of this approach is the significant time(1-10 s) per point, leading to unreasonable data acquisition times. A100×100 pixel requires a time on the order of 10s of hours.

Heretofore, the requirements of maximizing information about tip-surfaceinteractions and obtaining (a) independent amplitude, resonant frequencyand Q-factor parameters and (b) characterization of the completebehavior of the system referred to above have not been fully met. Whatis needed is a solution that solves all of these problems, preferablysimultaneously.

SUMMARY OF THE INVENTION

There is a need for the following embodiments of the invention. Ofcourse, the invention is not limited to these embodiments.

According to an embodiment of the invention, a process comprises:generation of the excitation signal having finite and predefinedamplitude and phase spectrum in a given frequency band(s); using thusgenerated excitation signal to excite SPM probe in the vicinity of thesample surface electrically, optically, mechanically, magnetically, orotherwise; measuring the mechanical or other response of the probe inthe predefined frequency range; analyzing the obtained data to extractrelevant dynamic parameters of probe behavior such as quality factor,resonant frequency, or full amplitude-frequency and phase-frequencycurve in a predefined range, that contain information on surfaceproperties. In a static band excitation embodiment, the excitationsignal is generated prior to imaging and is not changed from point topoint. In an adaptive band excitation embodiment, the excitation signalis re-synthesized at each point to accommodate the changes in dynamicbehavior of the probe in response to local surface properties, i.e.there is an active operational feedback.

According to another embodiment of the invention, a machine comprises: aprobe; a shielded sample holder with a transducer or other excitationmechanism; a signal generation circuit coupled to the transducer; adetector coupled to the probe; and a signal analysis component that iscapable of (a) fast determination of resonant frequency; a qualityfactor; and an amplitude, or (b) saving full amplitude/phase—frequencydata set at each point.

These, and other, embodiments of the invention will be betterappreciated and understood when considered in conjunction with thefollowing description and the accompanying drawings. It should beunderstood, however, that the following description, while indicatingvarious embodiments of the invention and numerous specific detailsthereof, is given by way of illustration and not of limitation. Manysubstitutions, modifications, additions and/or rearrangements may bemade within the scope of an embodiment of the invention withoutdeparting from the spirit thereof, and embodiments of the inventioninclude all such substitutions, modifications, additions and/orrearrangements.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawings accompanying and forming part of this specification areincluded to depict certain embodiments of the invention. A clearerconception of embodiments of the invention, and of the componentscombinable with, and operation of systems provided with, embodiments ofthe invention, will become more readily apparent by referring to theexemplary, and therefore nonlimiting, embodiments illustrated in thedrawings, wherein identical reference numerals (if they occur in morethan one view) designate the same elements. Embodiments of the inventionmay be better understood by reference to one or more of these drawingsin combination with the description presented herein. It should be notedthat the features illustrated in the drawings are not necessarily drawnto scale.

FIG. 1 is a view of a conventional a scanning probe microscope,appropriately labeled “PRIOR ART.”

FIGS. 2 a-2f show the characteristic excitation signals in the Fourier(top) and time (bottom) domains for a sinusoidal signal at signalfrequency, in frequency tracking, and in pulse modes, constituting thestate of the art in the field and appropriately labeled “PRIOR ART.”

FIGS. 3 a-3f show the frequency spectrums of examples of possible bandexcitation signals, representing embodiments of the invention.

FIGS. 4 a-4e show possible amplitude and phase distributions for bandexcitation signal in Fourier and time domains, representing embodimentsof the invention.

FIGS. 5 a-5d show further example of band excitation plots in time andfrequency domains, representing embodiments of the invention.

FIGS. 6 a-6f show more band excitation plots of different widths andpositions, representing embodiments of the invention.

FIGS. 7 a-7d show the effect of surface topography on contact stiffnessand as well as amplitude-frequency curves, representing an embodiment ofthe invention as applied to Piezoresponse Force Microscopy.

FIGS. 8 a-8c show PFM measurements with resonance enhancement andresonance spectra at selected locations within the domains and at thedomain wall of a sample, representing an embodiment of the invention.

FIGS. 9 a-9g show resonance enhanced PFM images obtained by the methodof the invention, representing an embodiment of the invention.

FIGS. 10 a-10 c show images generated by point-by-point band excitationscans, representing an embodiment of the invention.

FIGS. 11 a-11 d show the time evolution of the resonance structure ofthe cantilever on approaching and then withdrawing from the surface (2Dgraph); also shown are time dependences of amplitude, resonantfrequency, and Q-factor extracted from this data (the acquisition timeis˜100s), representing an embodiment of the invention.

FIG. 12 shows an apparatus, representing an embodiment of the invention.

DESCRIPTION OF PREFERRED EMBODIMENTS

Embodiments of the invention and the various features and advantageousdetails thereof are explained more fully with reference to thenonlimiting embodiments that are illustrated in the accompanyingdrawings and detailed in the following description. Descriptions of wellknown starting materials, processing techniques, components andequipment are omitted so as not to unnecessarily obscure the embodimentsof the invention in detail. It should be understood, however, that thedetailed description and the specific examples, while indicatingpreferred embodiments of the invention, are given by way of illustrationonly and not by way of limitation. Various substitutions, modifications,additions and/or rearrangements within the spirit and/or scope of theunderlying inventive concept will become apparent to those skilled inthe art from this disclosure.

Within this application several publications are referenced by Arabicnumerals, or principal author's name followed by year of publication,within parentheses or brackets. Full citations for these, and other,publications may be found at the end of the specification immediatelypreceding the claims after the section heading References. Thedisclosures of all these publications in their entireties are herebyexpressly incorporated by reference herein for the purpose of indicatingthe background of embodiments of the invention and illustrating thestate of the art.

This invention represents a novel approach for Scanning Probe Microscopy(SPM) based on the use of an excitation signal with controlled amplitudeand phase density in the finite frequency range, distinguishing it fromconventional approaches utilizing either excitation at a singlefrequency or spectroscopic measurements at each point. Similarband-excitation approaches can be used for other micro-electromechanicalsystems and cantilever-based sensor platforms having well-definedresonances or other well-defined regions of interest in responsespectrum.

Scanning probe microscopy (SPM) methods existing to date are universallybased on the detection of the signal under a constant (e.g. current inScanning Tunneling Microscopy, cantilever deflection in contact modeAtomic Force Microscopy) or a periodic (e.g. amplitude in intermittentcontact mode AFM or frequency shift in non-contact AFM) excitation.Generally, the sensitivity, resolution, non-invasiveness, andquantitativeness of SPM increases from constant to oscillatory tofrequency-tracking regimes, mirroring the development of the field inthe last two decades. In addition, a number of SPM techniques (e.g.Force Volume Imaging, Pulsed Force Mode and Molecular Recognition Mode)are based on the specially designed large-amplitude waveforms thatprobes different parts of force-distance curve to distinguish short-andlong range interactions.

In the case of oscillatory SPM modes, the frequency of the excitation iseither constant (amplitude or phase detection), or is adjusted using theappropriate feedback loop to maintain the system at the resonance. Thesignal at the selected frequency is measured using a lock-in amplifieror similar circuitry. In the variable frequency case, the frequency istracked typically using phase locked loop (PLL) based circuitry. Theamplitude of the modulation signal is either constant or is adjustedusing an additional feedback loop to establish a constant responsesignal amplitude.

The aforementioned SPM modes, including static, constant excitationfrequency and adjustable excitation frequency, now constitute themainstay of the field and are incorporated in virtually all commercialor homebuilt SPM systems. However, we note that the oscillatory SPMmodes to date are invariably limited to a single operational frequency(either zero for constant excitation, or selected frequency foroscillation). Hence, the measured response provides the information forsystem behavior only at the same frequency or its higher-orderharmonics, limiting the amount of information obtainable by SPM. Methodsbased on the direct sampling of the force-distance curve also belong tothis class, since either static force (force-distance measurements perse) or response at single frequency is measured at different positionsof the probe tip with respect to the surface.

Here, we note that the approach based on single-frequency measurementsis inherently limited in the amount of information on sample propertiesit can provide. As an example, in force-based SPMs the amplitude andquality factor (Q-factor) of the cantilever vibrating in contact withthe surface under constant mechanical excitation (e.g. in atomic forceacoustic microscopy, ultrasonic force microscopy, and force modulationimaging)¹ are two independent parameters containing the information onthe mechanical losses and elastic properties of the tip-surface system,from which real and imaginary parts of local indentation modulus of thematerial can be obtained. Note however, that in the single frequencyexcitation scheme, these parameters can not be separated, and dataanalysis based on simple harmonic oscillator model includesapproximations on the response at the resonance being inverselyproportional to the Q-factor.²

As a second example, in the electromechanical SPMs technique,Piezoresponse Force Microscopy,³ the response is strongly dependent bothon the Q actor and local electromechanical activity which varyindependently across the surface. These two contributions can not bedifferentiated by measurements at single frequency. In addition,conventional PLL feedback loops have limited applicability due to phasechanges between domains of opposite polarity, in which case the responsephase can not serve as a reliable feedback signal.⁴ As a result, thistechnique to date is limited to strongly piezoelectric materials, sincetraditional SPM approaches for resonance amplification of weak signal isinapplicable to PFM. Furthermore, no information on electromechanicallosses in the system due to domain growth and nucleation, molecularreorientation, etc. can be obtained

As a third example, in force-based SPMs the amplitude and quality factor(Q-factor) of the cantilever vibrating in the vicinity of the surfaceunder mechanical excitation (e.g. in intermittent contact mode AFM andnon contact AFM)⁵ are two independent parameters containing informationon the losses and elastic properties of the tip-surface system fromwhich local properties on the sub-10 nanometer, molecular, and atomiclevels can be obtained. However, in the single frequency excitationscheme (either using constant frequency, frequency tracking, orfrequency tracking with adjustable driving signal), these parameters cannot be separated, and complex data analysis methods based on theinterpretation of both phase and frequency signals have been developedassuming the constant driving force.^(6,7)

As a fourth example, response at the resonant frequency and losses ofthe electrically biased or magnetized tips provides information on localelectrical dissipation (related to e.g. carrier concentration andmobility)⁸ or magnetic dissipation (related to magnetization dynamics).⁹However, the data acquisition and analysis in these cases is subject tosame limitations as in examples¹⁻³.

Specifically, in all the methods above, the information, e.g. amount ofdissipated energy can be determined from the amplitude and phase at afixed frequency in constant frequency methods, or from amplitude atresonant frequency in frequency tracking methods, if and only if thedriving force acting on the system is constant. The typical exampleinclude position-independent mechanical driving on the cantilever usingpiezoactuator as embodied in intermittent contact AFM, phase AFM, orAtomic Force Acoustic Microscopy. In all cases when the driving forcedepends on electrostatic, magnetic, or electromechanical or otherposition-dependent forces, this is no longer the case. Moreover, thisderivation relies on the oscillations being perfectly sinusoidal, i.e.no higher harmonics of excitation frequency should be generated.

Finally, in all SPM techniques operating under the conditions in whichtip-surface interactions are non-linear, the amplitude-frequency andphase-frequency curves can differ significantly from the ideal harmonicoscillator or vibrating cantilever models, i.e. can includenonsymmetrical clipped main peaks, satellite peaks, etc. The shape ofthe response curve in the vicinity of resonance then containsinformation on the micromechanics and dynamics of tip-surfaceinteractions. However, this information can not be deduced frommeasurements at a single frequency.

This limitation was realized by many researchers working in thesefields, and a number of approaches based on the measurement ofamplitude-frequency curves at each point of the image were suggested.The methods to achieve this goal include for example: Sweeping theexcitation frequency with subsequent lock-in detection of the signal.However, this approach requires significant time (˜1-10 s) per point,since each frequency is sampled sequentially. This renders acquisitionof a high resolution image impractical, since required times for 100×100pixel images exceed 10s of hours.

Detection of the spectrum of thermal oscillation of cantilever orcantilever excited in the broad frequency band, by using spectrumanalyzer or similar methods. In this case, the full spectrum in eachpoint is obtained. However, this approach requires significant time ateach point due to low signal levels, (since excitation is performedsimultaneously at all frequencies), necessitating long acquisition timesat each pixel. In addition, phase information is lost in thesemeasurements.

Measuring the response to the step-function excitation and subsequentdetection of the cantilever response.¹⁰ In this case, the Fourierspectrum of excitation signal is constant, and this technique will alsosuffer from the limitations imposed by long acquisition times.

To summarize, in cases (a,b,c), the data acquisition at each point isextremely slow (>1 s), which renders imaging impractical. Moreover, dataprocessing is time consuming, and thus response can not be used as afeedback signal for fast imaging. We recognize that the limitations ofthese methods discussed above are imposed by the choice of the region ofthe Fourier space of the system probed during measurements. Singlefrequency techniques excite and sample only at a single frequency. Thisallows fast imaging and high signal levels, but the information on thefrequency-dependent response is lost. Spectroscopic techniques sampleall Fourier space (as limited by the bandwidth of the electronics used).However, the response amplitude is small (since excitation is performedsimultaneously at all frequencies), necessitating long acquisitiontimes; alternatively, the frequencies are probed sequentially at eachpoint, resulting in very long acquisition times

The invention can include obtaining information on the phase andamplitude of the probe (e.g. cantilever) response not only at a singlefrequency, but also in the finite frequency region around the chosenfrequency. As an example, mapping the amplitude- and phase-frequencybehavior in the vicinity of the resonance allows to determine resonancefrequency, quality factor, and amplitude independently. In a moregeneral case, exact knowledge of amplitude-and phase-frequency behaviorin the vicinity of the resonance provides a more complete description ofthe system in cases when single harmonic oscillator model isinapplicable due to non-linear interactions, et cetera. Furthermore, theinvention can include the use of thus determined parameters as afeedback signal to optimize imaging conditions. We also note that ofgreat practical interest are responses in selected regions of Fourierspace; and the invention can include obtaining information on amplitudeand phase behavior at resonances and in the vicinity of resonances, asopposed to either full spectrum, or response at a single or several(e.g. main signal and higher harmonics) selected frequencies.

Here, we propose the approach based on an adaptive digitally synthesizedsignal that excites multiple frequencies within selected frequency range(band of frequencies) simultaneously, avoiding the limitations of singleexcitation frequency (either constant or frequency-tracking) methods. Wedescribe the principles, implementation, and possible applications.

To determine the response of the cantilever in the selected frequencyinterval, e.g. in the vicinity of the resonance, the excitation signalhaving specified spectral density and phase content is defined. Thetraditional excitation signals are illustrated in FIGS. 2 a-2 f. Theconstant frequency is shown in FIGS. 2 a,b, and the frequency-trackingis shown in FIGS. 2 c,d, where feedback from the sample is measured andthe frequency is adjusted to keep the tip at resonance. The pulseexcitation is shown in FIGS. 2 e-2f. Shown are signals both in Fourierand time domains. These excitation schemes constitute the state of theart. Possible examples of signals in the band-excitation method areillustrated in FIGS. 3 a-3f and 4a-4 e, where the response in theselected frequency window around the resonance is excited. In FIG. 3 a,the excitation signal has a uniform spectral density. In FIG. 3 b, thespectral density on the tails ω₁ and ω₂ of the resonance peak isincreased to achieve better sampling. Thus the spectral densitydistribution of the resonant frequency band can take a parabolic shape.In FIG. 3 c, several resonance windows are excited simultaneously. InFIG. 3 d it is shown how the phase content of the signal can becontrolled to, for example, achieve Q-control amplification. FIGS. 3 e-3f illustrate two alternative signals. The excitation signal can beselected at the beginning of the imaging, or adapted at each point, sothat the center of the excitation window follows the resonance frequencyor the phase content is updated during the imaging.

In FIGS. 4 a-4 e we illustrate the effect of the phase content on BEsignal, including (FIG. 4 a) a signal with constant amplitude in a givenfrequency range and (FIG. 4 b) a signal with phase constant and (FIG. 4d) a signal with phase varied for optimal intensity. Also shown in FIGS.4 c and 4 e are corresponding signals in time domain. Finally, theexcitation waveform can be tailored in such a way as to provide anarbitrary system response (fully digital Q-control). In all cases, theexcitation signal can be synthesized prior to scan and kept constantduring imaging (static band excitation), or the excitation signal can becontinuously changed in response to sample properties (adaptive bandexcitation) as described below.

Thus synthesized signal is used as an excitation signal in the SPM,where the possible excitation schemes mirror those in conventional SPMand include, but are not limited to existing techniques such as

(a) mechanical excitation of the cantilever by the piezo actuator at thebase of the tip, or integrated in the cantilever(b) similar schemes with two or more actuators for excitation of e.g.torsional cantilever oscillations(c) electrical excitation by the bias applied to the cantilever abovethe surface to detect electrostatic forces(d) electromechanical excitation by the bias applied to the cantileverin contact with the surface to detect local piezoelectric properties(e) vertical, lateral, or longitudinal oscillator placed below thesample(f) excitation by magnetic coil(g) excitation by modulated light beam(h) excitation by electric current flowing through the cantilever(i) alternative force sensors based on e.g. membranes^(11,12)

Detected are the flexural and torsional responses of the cantilever, orequivalent displacement signals for alternative force sensors. Theresponse signal from photodiode, interferometer, capacitive,piezoresistive, or any other position sensor is read by the fast SPMelectronics and Fourier transformed to yield the frequency response ofthe system, i.e. amplitude and phase-frequency curves at each point.Alternatively, other integral transforms can be used to detectappropriate signal characteristics. Signal can be detected in thefrequency window corresponding to the excitation signal, orbroader/narrower window to e.g. detect second and higher order harmoniccomponents of response.

The microscope operation in signal acquisition can be performed in amanner

-   -   similar to force volume imaging, in which the tip approaches the        surface, the response at a single point is detected, and the tip        is shifted to second position    -   while continuously scanning the surface    -   in a spectroscopic mode, when the response is determined as a        function of probe-surface separation similar to force-distance        measurements, tip bias, or any other parameter    -   or any combination thereof.

The 3D data array obtained as described above is analyzed to yieldrelevant parameters of cantilever behavior. In the harmonic oscillatorapproximation, these including the resonance frequency or frequencies,response at the resonance frequency or frequencies, and correspondingQ-factors. Alternatively, statistical characteristics insensitive to themodel (e.g. statistical momentums of the amplitude-frequency response)or more complex analytical models can be used. Thus obtained parametersare stored as images and can be used as a feedback signal for microscopeoperation (static band excitation). Excitation signal can be synthesizedin each point to accommodate the change in sample properties (adaptiveband excitation) using feedback.

In the first approach, the band excitations signal is synthesized priorto the image acquisition and is not changed during the scanning. Theresponse, i.e. the array of amplitude-frequency and phase-frequencydata, is collected at each point and can be analyzed during imageacquisition or afterwards. In the former case, the selected responsesare stored as images, or used as feedback signals. The synthesizedsignal form is controlled to achieve e.g. the following specificfeatures (not limiting):

(a) achieve better sampling of the tails of resonance peak by increasingthe spectral density of input signal away from the resonance,(c) increase the effective Q-factor of the cantilever by selecting thephase content of excitation signal,(d) track several resonances simultaneously (e.g. for precisionmeasurements of contact stiffness and local elastic properties)(e) track several harmonics of a single resonance (e.g. for precisionmeasurements of cantilever oscillations modes). The band over which thesignal is collected and analyzed is not limited to the band offrequencies which was excited.

In the adaptive band excitation approach, the signal is analyzed at eachpoint of the image (e.g. using field programmable gate arrayelectronics) and obtained responses, including, but not limited to,resonant frequency, maximum response, and Q-factor, are either stored asimages, or used as feedback signals. The modulation signal is thensynthesized at each point to adjust for changes in local properties,e.g. shift in the resonant frequency. The possible implementations ofthe adaptive band excitation include, but are not limited to

(a) increase signal level by narrowing the frequency window around theresonance in such a manner that the center of resonance window followsthe resonant frequency of the cantilever(b) adapt the phase component of the signal to maximize the Q-factor,and hence the response, at each point.

One possible embodiment of the apparatus of the invention is illustratedin FIG. 12. A cantilever 900 measures surface-tip interaction with asample held in a sample holder 1208. A transducer 908 located within thesample holder and below the sample applies force to induce cantileverdeflections. Alternatively, the transducer can be located at thecantilever base or incorporated in the cantilever. The type oftransducer used depends on the applications, and can include, but not belimited to, a piezo actuator, voltage bias generator, magnetic coil,heating element, or a coherent light generator. The band excitationsignal generator circuit generates excitation waveform sent to the tip,actuator, or other driver. The detector circuit 1201 measures the timedependent deflections of the cantilever at each position of thecantilever as it scans across the sample surface. The analysis circuitanalyzes the probe response, and using a certain analysis procedure,including, but not limited to Fourier transform, generates anamplitude-frequency and phase-frequency curves for each pixel of thesample surface. The amplitude frequency and phase-frequency curves canbe analyzed in real time to extract resonant frequency, maximumamplitude, and Q factor in the simple harmonic oscillator approximation,set of relevant numerical parameters in alternative analytical models.The curves at each point can also be stored for subsequent analysis as a3D data set. The relevant characteristics (e.g. Q-factor, resonantfrequency, etc) can be used as a feedback signal to update the generatedwaveform in each point of image or spectrum step. Possibleimplementations of signal generation and analysis circuits can includehardware, such as integrated circuits or field programmable gate arrays,or through software programs on a computer.

The invention can be utilized in contexts other than scanning probemicroscopy. The invention can be utilized in the context of atomic forcemicroscopy. The invention can be utilized in the context of frictionalforce microscopy. The band excitation method can be applied to othernanomechanical devices based on cantilevers or other forms of resonantmechanical, or electromechanical, or electrical detection. Inparticular, the BE approach can be used for cantilever-based chemicaland biological sensors. In this case, changes in Q-factor and resonantfrequency detected simultaneously by BE approach will provideinformation on the chemical changes in the functional layer on thecantilever, indicative on the presence of specific agent. Anotherembodiment of BE method will be the micromechanical resonator circuitsbased on silicon membranes or more complex elements. Finally, BEapproach can be used for other types of SPM sensors, including recentlydeveloped membrane-based FIRAT and Delft sensors.

The detection and transformation of the real-time response of thecantilever produces an immense amount of data (millions of data pointsper pixel), as required by the Nyquist criterion (sampling rate shouldexceed twice the highest frequency component of the signal. Hence, toprobe dynamics at the MHz level, the sampling rate should be ˜2 MHz,corresponding to ˜1 Gb data array per standard 512×512 point image). Itis of critical importance that this data be managed, analyzed, anddistilled in such a way as to reduce it and extract only the relevantinformation (e.g., to around 10 parameters per pixel).

Fourier Transforms

The primary embodiment of the invention is based on the Fouriertransform, specifically discrete Fourier transform, to reduce the datafrom real time to frequency domain, with additional analysis techniquesto analyze the response in frequency domain. Due to linearity of Fouriertransform, the ratio of the amplitude content of excitation and responsesignal gives the frequency dependence of the amplitude of the systemresponse, and the difference between phase content of the response andexcitation signal yields the frequency dependence of the phase of thesystem response in the predefined frequency interval.

Linear Transforms

A similar approach can be based on other linear and more complexintegral transformations, including integral transforms of the kind

F(y, t) = ∫_(t)^(t + T)R(τ)G(y(t − τ)) τF(ω) = ∫_(t)^(t + T)R(t)G(ω, t) t

Where R(t) is the time-dependent response signal of the cantilever,G(ω,t) is the kernel of the transform, and F(ω) is the frequencydependence of the response signal. For Fourier transform, the kernel isG(ω,t)=e^(iαt)

The ultimate goal of response analysis in scanning probe microscopy isto determine not only the frequency content of the response, but toassociate that frequency information to a specific location. However,uncertainty principles dictate the inherent limitations that exist onthe amount or quality of frequency information one can ascertain from aspecific interval in time (or space in the case of scanning probe data).There are several techniques developed within the framework oftime-frequency and wavelet analysis that account for limitations posedby uncertainty and still provide useful information on the spatialvariation of the response signal frequency content. In BE signalanalysis it is necessary to convert the response signal of a singleline-scan to a spectrogram or (a 2-D amplitude vs. frequency and spacemap) or similar 2-D map. Time-frequency and wavelet analysis provide thetools to construct these plots. The most straight-forward time-frequencyanalysis technique is the ‘windowed-’ or ‘short time-’ Fouriertransform. In this, a ‘windowing’ function, H(τ−t), restricts thefrequency analysis to a specified interval of time (space)

F(ω, t) = ∫_(t)^(t + T)R(τ)G(ω, τ)H(τ − t) τG(ω, τ) = ^(− ω τ)

In the standard Fourier transform H(t−τ) is 1. The simplest window isthe rectangle function. Slightly more complex windows might include theGaussian, Hann, or Hamming windows.

Additional variability can be introduced to the windowing function tocontrol, for instance, its width so that

H(α, τ−t)

Wavelet Transforms

A subset of time-frequency analysis is wavelet analysis, in which atakes on a specific role in the widowing function as the dilationparameter

${H\left( {a,{\tau - t}} \right)} = {H\left( \frac{\tau - t}{a} \right)}$${W\left( {a,t} \right)} = {\int_{t}^{t + T}{{R(\tau)}{H\left( \frac{\tau - t}{a} \right)}\ {\tau}}}$

There are several commonly recognized and well-studied waveletfunctions:

Mexican hat or mother wavelet (pg. 6, eq. 2.1, P.S. Addison, 2002)Morlet wavelet (pg. 35, eq. 2.37, Addison)Haar wavelet (pg. 73, eq. 3.32, Addison)The Daubechies family of wavelets (pg 79, eq. 3.47, pg. 104,-116Addison)

Windowing functions and/or wavelets can be specially tuned to detectspecific events within the response signal. This makes them particularlyvaluable in scanning probe signal analysis in searching for particularvibration signatures that would indicate the presence or absence of anevent of interest (e.g. molecular binding, quaziparticle emission, etc).

Other Integral Transforms:

Other integral transforms can be used as well to generatespectrogram-like maps. Most notably the Wigner transform.

${F\left( {\omega,t} \right)} = {\int_{t}^{t + T}{{R^{*}\left( {t - {\frac{1}{2}\tau}} \right)}{R\left( {t + {\frac{1}{2}\tau}} \right)}{G\left( {\omega,\tau} \right)}\ {\tau}}}$G(ω, τ) = ^(− ω τ)

A general classification of integral transforms discussed above is asfollows (from pg 136, time-frequency analysis, L. Cohen, 1995)

${F\left( {\omega,t} \right)} = {\int{\int{\int{{{s^{*}\left( {u - {\frac{1}{2}\tau}} \right)} \cdot {s\left( {u + {\frac{1}{2}\tau}} \right)} \cdot {G\left( {\omega,\tau} \right)} \cdot ^{{{- }\; \theta \; t} - {\; \tau \; \omega} + {\; \theta \; u}}}{u}{\tau}{\theta}}}}}$

Where s and s^(•) are the response signal and the complex conjugate ofthe response signal, respectively, and G(ω,τ) is the kernel. Under thisclassification scheme, the kernel for the Wigner transform is 1. Thekernel for the windowed Fourier transform is

${G\left( {\omega,\tau} \right)} = {\int{{{H^{*}\left( {u - {\frac{1}{2}\tau}} \right)} \cdot ^{{- }\; \theta \; t} \cdot {H\left( {u + {\frac{1}{2}\tau}} \right)}}{u}}}$

Other kernels include:

Margenau-Hill, cos(1/2θτ)Kirkwood-Rihaczek, e^(iθτ/2)Born-Jordan (sinc), sin (αθτ)/αθτPage, e^(i)θ|τ|Choi-Williams, e^(−θ) ² ^(τ) ² ^(/σ)

Other transform procedures which are particularly useful for improvingthe frequency resolution over a specific frequency range of discretesignals include the Chirp Fourier Transform (pg. 151, A course inDigital Signal Processing, B. Porat, 1997) and the Zoom FFT (pg. 153, Acourse in Digital Signal Processing, B. Porat, 1997).

Other forms of data analysis may include analysis of time series at eachpoint to extract (e.g. Lyapunov) exponents from chaotic data. Otherforms of data analysis may also include single event statisticalanalysis (e.g. density of events, etc).

The invention can include other forms of analysis. The measuredcantilever response either in time domain, Fourier domain, and phasespace of other integral transform can be fitted to a specific model toreduce the data array to a small number model-specific parameters. Oneembodiment of the invention includes fit to the simple harmonicoscillator model Eq. (1), where resonant frequency, maximum response,and Q-factor provide complete description of the system dynamics.

The other form of analysis include fitting to the solution of beamequation or any other linear differential equation describing probedynamics, for which the solution can be represented as a linearsuperposition of partial solutions with coefficients dependent on localproperties. As an example, in a specific case of the cantilevervibrating under the effect of local and distributed electros tic forcesand surface displacement in Piezoresponse force microscopy, thecantilever dynamics in the absence of damping is given by

${\frac{^{4}u}{x^{4}} + {\frac{\rho \; S_{c}}{EI}\frac{^{2}u}{t^{2}}}} = \frac{q\left( {x,t} \right)}{EI}$

where E is the Young's modulus of cantilever material, I is the momentof inertia of the cross-section, ρ is density, S_(c) is cross-sectionarea, and q(x,t) is the distributed force acting on the cantilever. Fora rectangular cantilever S_(c)=wh and I=wh³/12, where w is thecantilever width and h is thickness. The cantilever spring constant, k,is related to the geometric parameters of the cantilever byk=3EI/L³=Ewh³/4L³. In beam-deflection AFM, the deflection angle of thecantilever, θ, is measured by the deflection of the laser beam at x=L,and is related to the local slope as θ=arctan(u′(L))≈u′(L). For a purelyvertical displacement, the relationship between cantilever deflectionangle and measured height is A=2θL/3. This equation provides therelationship between cantilever deflection induced by longitudinal orelectrostatic interactions and detected vertical PFM signal.

Eq. (1) is solved in the frequency domain by introducingu(x,t)=u₀(x)e^(iωt), q(x,t)=q₀e^(iωt, where u) ₀ is the displacementamplitude, q₀ is a uniform load per unit length, t is time, and ω ismodulation frequency. After substitution, Eq. (1) is:

$\begin{matrix}{{{\frac{^{4}u_{0}}{x^{4}}\kappa^{4}u_{0}} + \frac{q_{0}}{EI}},} & (2)\end{matrix}$

where κ⁴=ω² ρS_(c)/EI is the wave vector. On the clamped end of thecantilever, the displacement and deflection angle are zero, yielding theboundary conditions

u₀(0)=0 and u₀(0)=0,  (3a,b)

On the supported end, in the limit of linear elastic contact theboundary conditions for moment and shear force are

EIu₀″(L)=k₂H(d₂−u₀′(L)H) and EIu₀′″(L)=−ƒ₀+k₁(u₀(L)−d₁)  (4a,b)

where ƒ₀ is the first harmonic of the local force acting on the tip. Fornon-piezoelectric materials, d₁=d₂0, while for zero electrostatic force,ƒ₀=0, providing purely electromechanical and electrostatic limitingcases for Eq. (4).

After solving the linear Eq. (2) and using EI=kL³/3, the deflectionangle is

$\begin{matrix}{{{\theta (\beta)} = \frac{{{A_{v}(\beta)}d_{1}} + {{A_{l}(\beta)}d_{2}} + {{A_{e}(\beta)}f_{0}} + {{A_{q}(\beta)}q_{0}}}{N(\beta)}}{where}} & (5) \\{{A_{v}(\beta)} = {3\; \beta^{4}k_{1}{kL}\; \sin \; \beta \; \sinh \; \beta}} & (6) \\{{A_{l}(\beta)} = {3\; \beta^{2}{{Hk}_{2}\begin{pmatrix}{{3\; k_{1}} + {\cosh \; \beta \left( {{{- 3}\; k_{1}\cos \; \beta} + {\beta^{3}k\; \sin \; \beta}} \right)} +} \\{\beta^{3}k\; \cos \; \beta \; \sinh \; \beta}\end{pmatrix}}}} & (7) \\{{A_{e}(\beta)} = {3\; \beta^{4}{kL}\; \sin \; \beta \; \sinh \; \beta}} & (8) \\{{A_{q}(\beta)} = {3\; {L^{2}\begin{pmatrix}{{3\; {k_{1}\left( {{\cos \; \beta} - {\cosh \; \beta}} \right)}} - {k\; \beta^{3}\sin \; \beta} +} \\{\left( {{k\; \beta^{3}} + {3\; k_{1}\sin \; \beta}} \right)\sinh \; \beta}\end{pmatrix}}}} & (9) \\{{N(\beta)} = {\beta^{2}\left( {\begin{matrix}{{9\; H^{2}k_{1}k_{2}} + {\beta^{4}k^{2}L^{2}} +} \\{\cosh \; {\beta \begin{pmatrix}{\left( {{{- 9}\; H^{2}k_{1}k_{2}} + {\beta^{4}k^{2}L^{2}}} \right)\cos \; {\beta++}} \\{3\beta \; {k\left( {{k_{1}L^{2}} + {H^{2}k_{2}\beta^{2}}} \right)}\sin \; \beta}\end{pmatrix}}} \\{3\; \beta \; {k\left( {{{- k_{1}}L^{2}} + {H^{2}k_{2}\beta^{2}}} \right)}\cos \; \beta \; \sinh \; \beta}\end{matrix} +} \right)}} & (10)\end{matrix}$

and the dimensionless wave number is β=κL. In this specific case,fitting of the cantilever response allows determination of verticalpiezoelectric coefficient, d₁, longitudinal piezoelectric coefficient,d₂, local electrostatic force, ƒ₀, and distributed electrostatic force,q, at each point. Similar approaches can be applied to more complexdifferential equations containing intrinsic and tip-surface dampingterms, etcetera. Still more complex forms of analysis include lineardifferential equations with distance-dependent forces including bothconservative and dissipative components, for which solutions can befound for a small number explicitly or parametrically defined modelspecific parameters.

The invention can include the use of characterization data to controlthe band excitation signal. The model specific parameters describingsystem dynamics can be recorded as SPM images or be used as an input tomodify the excitation signal. The simplest form of such modificationinclude changing the frequency interval (the position of each edge ofthe band). Other possible forms include:

vary phase contentposition of the phase cross-over as in Q-controlcombine (add, subtract, multiply) the response signal to excitationsignal to artificially either increase or decrease the effectiveQ-factor of the systemvary amplitude density to achieve optimal sampling of region of interest

The signal can be used as a feedback for updating the waveform forproperty measurements, or for topographic imaging.

EXAMPLES

Specific embodiments of the invention will now be further described bythe following, nonlimiting examples which will serve to illustrate insome detail various features. The following examples are included tofacilitate an understanding of ways in which an embodiment of theinvention may be practiced. It should be appreciated that the exampleswhich follow represent embodiments discovered to function well in thepractice of the invention, and thus can be considered to constitutepreferred mode(s) for the practice of the embodiments of the invention.However, it should be appreciated that many changes can be made in theexemplary embodiments which are disclosed while still obtaining like orsimilar result without departing from the spirit and scope of anembodiment of the invention. Accordingly, the examples should not beconstrued as limiting the scope of the invention.

Example 1 Band Excitation in Time and Frequency Plots

Example 1 illustrates the concept of a static band excitation applied toa piezo-actuator with response spectrum. Excitation waveform forexcitation bands of different widths and positions and response spectrafor these different waveforms are shown.

This method differs from the method using lock-in and sweeps in that asingle complex waveform is used to excite a continuous band offrequencies instead of just driving the system with a single frequency.The results in FIG. 5 a were recorded using lock-in/sweep method andrepresent the amplitude of cantilever response as a function of dc tipbias and excitation frequency. The oscillations are induced by thepiezoelectric actuator placed below the tip. The FIG. 5 b shows thespectral amplitude response for a tip bias of 0 V. The frequency rangeis from 5000 Hz to 50,000 Hz. The tip used in these tests has a resonantfrequency of 12 kHz according to the manufacturer. The acquisition timefor data in FIGS. 5 a-5 b are ˜30 min (˜1s/frequency) and are limited bythe time required for the lock-in amplifier to switch to a differentfrequency.

Illustrated in FIG. 5 c are the results from the band excitationapproach. FIG. 5 c shows the variation in amplitude as a function oftime. The x-axis is in units of micro-seconds. So the duration of theplot is approximately 1/80^(th) of a second. The bottom line is thespecially chosen input function used to drive the piezo actuator. Thetop line is the vertical response of the tip as measured by the opticaldetector. No noise filtering system was used in any of theseexperiments. Note: the y-axis scale is not the same for the two datasets. These data were captured simultaneously using the data acquisitionsystem running at 100 ksamples/sec for each channel.

FIG. 5 d shows a plot of the Fourier transform of each of the signalsabove. The square spectrum shows the band of driving frequencies which,for this experiment, was chosen to span from 5 kHz to 40 kHz. The lowerline is the response spectrum of the piezo/tip system. Comparison ofthis plot with the results from the lock-in shown in FIGS. 3 a-3 breveals some similarities and some differences. Many of the peaks arelocated at the same frequencies and many of the interesting features arethe same in each image. However, the relative intensities of thefeatures are different. This discrepancy is probably due tonon-linearities in the cantilever response, when higher-order harmonicscontribute to the signal.

The plots in FIGS. 6 a-6 f show results from using excitations bands ofdifferent widths and positions. FIGS. 6 b, 6 d, and 6 f show theamplitude plots and FIGS. 6 a, 6 c, and 6 e the corresponding Fouriertransforms. As expected, there is almost no measurable response outsidethe excitation window. In this experiment we have excited just the bandsaround the resonant frequency. Note that the response curves do notchange appreciably on the narrowing of the frequency window.

Example 2 BE-PFM

Example 2 details the application of the static band excitationprinciple for piezoresponse force microscopy. In this example, theband-excitation method is used to perform resonance-enhancedPiezoresponse Force Microscopy.

Piezoresponse Force Microscopy and spectroscopy of domain and switchingdynamics at small excitation voltages requires resonance enhancement ofsmall surface displacements. The contact stiffness depends strongly onlocal elastic properties and topography, resulting in significantvariations of the resonant frequency. Moreover, electromechanicalresponse at the resonance is determined both by the local Q-factor andelectromechanical activity. Here we develop an approach forresonance-enhanced PFM that allows mapping of the localelectromechanical activity, contact stiffness, and loss factor, thusavoiding limitations inherent to conventional frequency-tracking. Weanticipate that resonance-enhanced PFM will be important for imagingweakly piezoelectric materials and probing inelastic phenomena inferroelectrics.

Piezoelectric coupling between electrical and mechanical phenomena isextremely common in inorganic materials (20 out of 32 symmetry groupsare piezoelectric)¹³ and is nearly universal in biopolymers such asproteins and polysaccharides.¹⁴ In the last decade, Piezoresponse ForceMicroscopy (PFM) has emerged as a key tool for electromechanicalimaging, polarization control, and local spectroscopic measurements inferroelectric materials on the nanoscale.^(15,16,17) To date, the vastmajority of PFM studies has been performed on ferroelectric materialswith relatively strong (d˜20-2000 pm/V) piezoelectric coefficients.However, PFM imaging and spectroscopy of domain dynamics at smallexcitation voltages and imaging of weak (d˜1-10 pm/V) piezoelectricmaterials such as III-V nitrides¹⁸ and biopolymers¹⁹ necessitate theamplification of the response signal compared to the amplitude ofsurface oscillations. A number of groups have suggested to increasedetection limits in PFM and decrease modulation voltages by imaging atfrequencies corresponding to the contact resonances of thecantilever.^(20,21,22) However, the contact resonances are extremelysensitive to the contact stiffness of the tip-surface junction due tosurface curvature and variation in local mechanical properties,²³ asillustrated in FIGS. 7 a-7 c. This effect is particularly pronounced fortopographic inhomogeneities on the length scale of contact radius suchas step edges, etc, which can be erroneously interpreted as PFMcontrast. FIG. 7 a shows how contact stiffness depends strongly onsurface topography through the variation of contact area and localmechanical properties. FIG. 7 b shows how the amplitude at a constantfrequency provides information on electrochemical activity on ahomogeneous surface. In contrast, FIG. 7 c shows how on inhomogeneoussurfaces, changes in contact resonant frequency can result in strongvariations in the signal, even on piezoelectrically uniform surfaces.

Here, we analyze the mechanisms for resonance enhancement in PFM anddevelop an approach for local electromechanical imaging based on therapid (10-100 ms/pixel) measurement of the amplitude vs. frequencyresponse curve at each point. This technique allows mapping of localelectromechanical activity, contact resonant frequency, and Q-factor,thereby deconvoluting the mechanical and electromechanical effects.Resonance-enhanced PFM measurements are demonstrated on polycrystallinelead titanate-zirconate (PZT) ceramics.

A detailed analysis of cantilever dynamics in PFM²⁴ shows that the PFMsignal is a linear combination of local and non-local electrostatic andlocal electromechanical contributions. The resonant frequencies aredetermined only by the weakly voltage dependent mechanical properties ofthe system and are independent of the relative contributions of theelectrostatic and electromechanical interactions. As shown by Sader,²⁵in the vicinity of the resonance for small damping (Q>10), theamplitude-frequency response can be described using the harmonicoscillator model to yield²⁶

$\begin{matrix}{{A_{i}(\omega)} = {\frac{A_{i}^{\max}{\omega_{i\; 0}^{2}/Q_{i}}}{\sqrt{\left( {\omega_{i\; 0}^{2} - \omega^{2}} \right)^{2} + \left( {{\omega\omega}_{i\; 0}/Q_{i}} \right)^{2}}}.}} & (1)\end{matrix}$

where A_(i) ^(max) is the signal at the frequency of i^(th) resonanceω_(i0) and Q₁ is the quality factor that describes energy losses in thesystem. The frequencies at which the system is most sensitive tocrosstalk occurs not at the resonance peak, but at the frequencies toeither side of the peak, ω_(im) ^(±), where the slope of the A(ω) curveis greatest. By using (ω_(i0)±ω_(im) ^(=)/ω) _(i0)=0.35/Q_(i), thechange in amplitude due to a shift in the resonant frequency is givenby:²⁷

$\begin{matrix}{{\delta \; A_{i}} = {\frac{4}{3\sqrt{3}}\frac{Q_{i}}{\omega_{i\; 0}}A_{i}^{\max}\delta \; {\omega_{i\; 0}.}}} & (2)\end{matrix}$

Variations in the contact resonant frequency, δω_(i0), due to the changein tip-surface contact stiffness can be obtained using analysis byMirman et al. The resonant frequencies of the cantilever are ω_(i)²=EIμ_(i) ⁴/mL⁴=μ_(i) ⁴k/3mL, where E is the Young's modulus of thecantilever material, l is the 2^(nd) moment of inertia of thecross-section, and m is mass per unit length. The dimensionless wavenumber, μ_(i), is related to the cantilever spring constant, k=3 El IL³,and tip-surface contact stiffness,

$\begin{matrix}{{k_{1},{as}}{{\mu_{i} = \frac{a_{i} + {b_{i}y_{1}}}{1 + {c_{i}y_{1}}}},}} & (3)\end{matrix}$

where Y₁=k₁/k and coefficients ai, bi, and ci for i-th resonance aregiven in Table I.

From Eqs. (2,3), the variation in the resonance-enhanced PFM signal dueto variations in the surface topography or local elastic properties is

TABLE 1 Dependence of the eigen frequency on resonance number (4)${\delta \; A_{i}} = {\frac{S}{3\sqrt{3}}Q_{i}A_{i}^{\max}\frac{b_{i} - {a_{i}c_{i}}}{\left( {1 + {c_{i}\gamma_{1}}} \right)\left( {a_{i} - {b_{i}\gamma_{1}}} \right)}{{\delta\gamma}_{1}.}}$Resonance a_(i) b_(i) c_(i) d_(i) f_(i) γ_(1c) 1 1.88 0.708 0.180 0.3455.10 3.83 2 4.69 0.145 0.0207 0.0630 98.7 39.7 3 7.85 0.0564 0.005550.0242 605 158 n $\pi \left( {n - \frac{1}{2}} \right)$$\frac{4.8}{\pi^{2}n^{2}}\left( {1 + \frac{1}{4n}} \right)$$\frac{4.8}{\pi^{3}n^{3}}$$\frac{1.6}{\pi^{3}n^{3}}\left( {{4n} + 1} \right)$$\frac{\pi^{3}n^{3}}{7.2}\left( {{2n} + 1} \right)$$\frac{\pi^{3}n^{3}}{4.8}\left( {1 - \frac{3}{8n}} \right)$Robust PFM imaging requires that changes in the signal due to shifts inthe resonant frequency, δA_(i), is small compared to the PFM signal,i.e. δA_(i)<αA_(i) ^(max), where constant α can be selected as α=0.1 orbelow, corresponding to an elastic cross-talk with the PFM signals of10% or below. The conditions for the variation in spring constant canthen be obtained from Eq. (4) in a straightforward manner. Inparticular, for high contact stiffnesses (Y₁>>1) the condition forrobust electromechanical imaging is δγ₁<3√{square root over (3)} αγ₁²d_(i)/(δQ), where the constant d_(i)=b_(i)c_(i)/(b_(i)−a_(i)c_(i)) islisted in Table I. Under typical PFM imaging conditions, k=4 N/m,k1=1000 N/m, and Q=20, the condition is δk₁ lk₁<0.27 for the firstresonance and δk₁lk₁<0.052 for the second. Note that while for the firstresonance, a 10% variation in the contact stiffness will not result in asignificant topographic cross-talk. However, this is no longer true forsecond and higher-order resonances.

For soft contact (Y₁<<1), corresponding to stiff cantilevers or highfrequencies at which inertial stiffening effects become important, thecondition for robust electromechanical imaging is δγ₁<3√{square rootover (3)}αƒ_(i)/(δQ), where constants f_(i)=a_(i)l(b_(i)=a_(i)c_(i)) arelisted in Table I. The cross-over between these regimes occurs forγ_(1c)=√{square root over (α_(i)/(b_(i)c_(i)))}, as listed in Table I.For the parameters above, this corresponds to i≧4 and the condition forrobust imaging is δk₁/k₁<0.005 for the 4th resonance and δk₁/k₁<0.05 forthe 5^(th) resonance. Note however that under such conditions,transduction of the displacement from the surface to the tip would bereduced.²⁴

From this analysis, it is clear that even small variations in elastic ortopographic conditions at the surface will result in strong cross-talkin resonance-enhanced PFM. Practically, resonance enhancement using aconstant excitation frequency for PFM can be used reliably only whenimaging anti-parallel ferroelectric domains on topographically uniformsurface, i.e. when elastic properties are uniform.

Successful resonance-enhanced PFM imaging on topographically orelastically inhomogeneous materials thus requires continuous tracking ofthe local contact resonant frequency. However, traditional phase-lockedloop based frequency-tracking schemes are sensitive to the phase of thesignal, as shown in FIG. 7 d. The phase of the response is opposite forantiparallel domains and changes by 180° across the resonance. However,due to the 180° phase change in the response with respect to the drivingsignal between antiparallel domains, the feedback loop becomes unstablefor domains of certain polarity. Moreover, the measured response at theresonant frequency depends on both the local electromechanical activityand Q-factor. Hence, local losses can not be determined unambiguouslyfrom the maximal signal strength as is the case for Atomic ForceAcoustic Microscopy²⁸ since Q-factor and piezoelectric activity can varyindependently. Thus, we have developed an approach forresonance-enhanced PFM based on the acquisition of anamplitude-frequency curve at each point.

PFM and R-PFM are implemented on a commercial Scanning Probe Microscopy(SPM) system (Veeco MultiMode NS-IIIA) equipped with additional functiongenerators. A custom-built, shielded sample holder was used to bias thetip directly thus avoiding cross-talk with the SPM electronics.Measurements were performed using Au coated tips (NSC-36 B, Micromasch,resonant frequency˜155 kHz, spring constant k˜1.75 N/m). To implementR-PFM, the microscope was configured in a manner similar to the forcevolume mode, shown in FIG. 6 a (experimental set-up for PFM measurementswith resonance enhancement). The tip approaches the surface verticallyin contact mode until the deflection set-point is achieved. Theamplitude-frequency data is then acquired at each point. Afteracquisition, the tip is moved to the next location and continued until amesh of evenly spaced M×N points is acquired. The A(ω) curves arecollected at each point as a 3D data array, with the typical acquisitiontime of 10-100 ms per point. The data is fitted to Eq. (1) to yieldcoefficients giving the local electromechanical response A_(i) ^(max),resonant frequency ω_(/0), and Q-factor. These coefficients are thenplotted as 2D maps. An example of a PFM image and A(ω) curves at threedifferent locations is shown in FIGS. 8 b and 8 c, respectively. FIG. 8b shows a map of the resonant PFM amplitude signal A_(i) ^(max) across aPZT surface. FIG. 8 c shows the response spectra and fits using Eq. (1)at selected locations within the domains and at the domain wall. Notethat the variation in the local resonant frequency between dissimilardomains is on the order of 10-30 kHz—an amount significantly exceedingthe width of the resonance peak, and that the position of the resonancepeak is significantly shifted and its magnitude is reduced at the domainwall.

Resonance-enhanced PFM images of polycrystalline PZT are shown in FIGS.9 a-9 g. The topographic image in FIG. 9 a shows topographic changes atthe grain boundary and features associated with preferential domainetching. The PFM amplitude and phase images at 2.0 MHz, i.e. far fromthe resonance, detail the local domain structure. Note in FIG. 9 b, (I)the strong enhancement of the PFM signal at the grain boundary, (II) thepresence of 180° domain walls within the grains, and (III) the regionwith zero response corresponding to an in-plane domain or embeddednon-ferroelectric particle. FIGS. 9 c and 9 d show a map of whereconvergence of the fitting procedure was successful (white) andunsuccessful (black). Unsuccessful convergence occurs where the responsesignal is too weak (e.g. region II of FIG. 9 b) and no resonance peak isdetectable. The maximal electromechanical response A_(i) ^(max) activitymap at the local resonance frequency in FIG. 9 e illustrates variationsin the piezoelectric response between the domains. Note that there is nosignificant enhancement of the response at the grain boundary, whilethere is a significant 150 kHz shift in the local resonant frequency asshown in FIG. 9 f. Finally, a map of the Q-factor in FIG. 9 gillustrates that losses are almost uniform throughout the sample. It isimportant to note that there are significant variations between theseimages, thus signifying that they provide complementary data about thelocal material properties. It is also important to note the goodagreement between FIG. 9 b and FIG. 9 e, which is indicative of theveracity of the electromechanical data.

To summarize, we have shown how to analyze the contrast formationmechanism in resonance-enhanced PFM. It is shown that even smallvariations in topography or local elastic properties result insignificant cross-talk in the PFM signal, hence limiting the constantfrequency resonance enhanced mode to the special case of antiparalleldomains on topographically uniform surface. The resonance-enhanced PFMdeveloped by the method of the invention allows real-space mapping ofelectromechanical activity, contact resonant frequency, and Q-factor,thus providing comprehensive information on the local mechanical andelectromechanical properties and avoiding the limitations of traditionalPLL-based frequency tracking feedback schemes. This approach will pavethe way for future high resolution studies of electromechanical activityin weakly piezoelectric materials and inelastic processes associatedwith domain nucleation and domain wall motion in ferroelectrics.

Example 3 Imaging Mechanical Properties

Example 3 details the application of static band-excitation principle toAtomic Force Acoustic Microscopy. Here, the band excitation set-up isused to image mechanical properties. The electrical bias applied to thetip induces electrostatic interactions between the tip and thesubstrate. These electrostatic interactions drive the cantilever. Thedynamic response of the cantilever is determined in part by themechanical stiffness of the tip surface junction. Therefore variationsin the position, height, and width of the peak of theamplitude-frequency response curve are directly related to variations inhardness across the substrate surface.

The results shown in FIGS. 10 a-10 c were obtained by a point-by-pointband excitation scan of a non-piezoelectric inter-metallic alloy(NiAlMo). This sample contains inclusions of a softer alloy within aharder matrix. The resonance frequency map shows the location of thesofter inclusions as regions with a lower resonant frequency (blue)within the harder matrix. In this case, the band excitation signal wassent directly to the tip. However, it is possible (and perhaps evenadvantageous) to send this same signal to a piezoelectric acoustic stagebelow the sample for operation in band excitation AFAM.

Example 4 Dissipation-Distance Curve

Example 4 details the application of static band-excitation principle toForce-distance measurements. Here, the band excitation set-up is used todetect the changes in cantilever response as a function of tip surfaceseparation. The electrical bias applied to the tip induces electrostaticinteractions between the tip and the substrate. These electrostaticinteractions drive the cantilever. The dynamic response of thecantilever is determined in part by the mechanical stiffness of thesystem, which is the sum of cantilever stiffness and the stiffness oftip surface junction. The data in FIG. 11 illustrates the change in theresonance spectrum of the system on approaching the surface (signalincreases due to increase in capacitive forces), on contact (signaldecreases due to increase in tip-surface junction), and on retraction oftip from the surfaces. Note the significant change of the resonantstructure from free to surface-bound cantilever. The correspondingforces, Q-factors, and resonant frequencies are plotted in FIGS. 11 b,11 c and 11 d and are extracted from data in FIG. 11 a using simpleharmonic oscillator fit.

The 4 examples illustrated should not be construed as the only suitableembodiments of the invention. The present invention can be applied toany scanning probe microscopy apparatus or methods, including, but notlimited to atomic force microscopy (AFM), electrostatic force microscopy(EFM), force modulation microscopy (FMM), kelvin probe force microscopy(KPFM), magnetic force microscopy (MFM), magnetic dissipation forcemicroscopy, magnetic resonance force microscopy (MRFM), near-fieldscanning optical microscopy (NSOM), scanning near-field opticalmicroscopy (SNOM), photon scanning tunneling microscopy (PSTM), scanningelectrochemical microscopy (SECM), scanning capacitance microscopy(SCM), scanning gate microscopy (SGM), scanning thermal microscopy(SThM), scanning tunneling microscopy (STM), and scanning voltagemicroscopy (SVM), as well as other embodiments of scanning probemethods. This BE method is also applicable to micro- and nanomechanicalsystems, including cantilever sensors, membrane sensors, etc. Finally,current flowing from tip to the surface can be measured in a similarfashion.

Definitions

The term program and/or the phrase computer program are intended to meana sequence of instructions designed for execution on a computer system(e.g., a program and/or computer program, may include a subroutine, afunction, a procedure, an object method, an object implementation, anexecutable application, an applet, a servlet, a source code, an objectcode, a shared library/dynamic load library and/or other sequence ofinstructions designed for execution on a computer or computer system).The phrase radio frequency is intended to mean frequencies less than orequal to approximately 300 GHz as well as the infrared spectrum.

The term substantially is intended to mean largely but not necessarilywholly that which is specified. The term approximately is intended tomean at least close to a given value (e.g., within 10% of). The termgenerally is intended to mean at least approaching a given state. Theterm coupled is intended to mean connected, although not necessarilydirectly, and not necessarily mechanically. The term proximate, as usedherein, is intended to mean close, near adjacent and/or coincident; andincludes spatial situations where specified functions and/or results (ifany) can be carried out and/or achieved. The term deploying is intendedto mean designing, building, shipping, installing and/or operating.

The terms first or one, and the phrases at least a first or at leastone, are intended to mean the singular or the plural unless it is clearfrom the intrinsic text of this document that it is meant otherwise. Theterms second or another, and the phrases at least a second or at leastanother, are intended to mean the singular or the plural unless it isclear from the intrinsic text of this document that it is meantotherwise. Unless expressly stated to the contrary in the intrinsic textof this document, the term or is intended to mean an inclusive or andnot an exclusive or. Specifically, a condition A or B is satisfied byany one of the following: A is true (or present) and B is false (or notpresent), A is false (or not present) and B is true (or present), andboth A and B are true (or present). The terms a or an are employed forgrammatical style and merely for convenience.

The term plurality is intended to mean two or more than two. The termany is intended to mean all applicable members of a set or at least asubset of all applicable members of the set. The phrase any integerderivable therein is intended to mean an integer between thecorresponding numbers recited in the specification. The phrase any rangederivable therein is intended to mean any range within suchcorresponding numbers. The term means, when followed by the term “for”is intended to mean hardware, firmware and/or software for achieving aresult. The term step, when followed by the term “for” is intended tomean a (sub)method, (sub)process and/or (sub)routine for achieving therecited result.

The terms “comprises,” “comprising,” “includes,” “including,” “has,”“having” or any other variation thereof, are intended to cover anon-exclusive inclusion. For example, a process, method, article, orapparatus that comprises a list of elements is not necessarily limitedto only those elements but may include other elements not expresslylisted or inherent to such process, method, article, or apparatus. Theterms “consisting” (consists, consisted) and/or “composing” (composes,composed) are intended to mean closed language that does not leave therecited method, apparatus or composition to the inclusion of procedures,structure(s) and/or ingredient(s) other than those recited except forancillaries, adjuncts and/or impurities ordinarily associated therewith.The recital of the term “essentially” along with the term “consisting”(consists, consisted) and/or “composing” (composes, composed), isintended to mean modified close language that leaves the recited method,apparatus and/or composition open only for the inclusion of unspecifiedprocedure(s), structure(s) and/or ingredient(s) which do not materiallyaffect the basic novel characteristics of the recited method, apparatusand/or composition.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. In case of conflict, thepresent specification, including definitions, will control.

CONCLUSION

The described embodiments and examples are illustrative only and notintended to be limiting. Although embodiments of the invention can beimplemented separately, embodiments of the invention may be integratedinto the system(s) with which they are associated. All the embodimentsof the invention disclosed herein can be made and used without undueexperimentation in light of the disclosure. Although the best mode ofthe invention contemplated by the inventor(s) is disclosed, embodimentsof the invention are not limited thereto. Embodiments of the inventionare not limited by theoretical statements (if any) recited herein. Theindividual steps of embodiments of the invention need not be performedin the disclosed manner, or combined in the disclosed sequences, but maybe performed in any and all manner and/or combined in any and allsequences. The individual components of embodiments of the inventionneed not be formed in the disclosed shapes, or combined in the disclosedconfigurations, but could be provided in any and all shapes, and/orcombined in any and all configurations. The individual components neednot be fabricated from the disclosed materials, but could be fabricatedfrom any and all suitable materials.

It can be appreciated by those of ordinary skill in the art to whichembodiments of the invention pertain that various substitutions,modifications, additions and/or rearrangements of the features ofembodiments of the invention may be made without deviating from thespirit and/or scope of the underlying inventive concept. All thedisclosed elements and features of each disclosed embodiment can becombined with, or substituted for, the disclosed elements and featuresof every other disclosed embodiment except where such elements orfeatures are mutually exclusive. The spirit and/or scope of theunderlying inventive concept as defined by the appended claims and theirequivalents cover all such substitutions, modifications, additionsand/or rearrangements. The appended claims are not to be interpreted asincluding means-plus-function limitations, unless such a limitation isexplicitly recited in a given claim using the phrase(s) “means for”and/or “step for.” Subgeneric embodiments of the invention aredelineated by the appended independent claims and their equivalents.Specific embodiments of the invention are differentiated by the appendeddependent claims and their equivalents.

REFERENCES

1. U. Rabe, Atomic Force Acoustic Microscopy, in Applied Scanning ProbeMethods, Vol II, Eds. B. Bhushan and H. Fuchs, Springer, N.Y. (2006).2. K. Yamanaka, Y. Maruyama, T. Tsuji, and K. Nakamoto, Appl. Phys.Lett. 78, 1939 (2001).3. Nanoscale Characterization of Ferroelectric Materials, ed. M. Alexeand A. Gruverman, Springer (2004).4. S. Jesse, B. Mirman, and S. V. Kalinin, Appl. Phys. Lett. 89, 022906(2006).5. R. Garcia and R. Pérez, Surf. Sci. Reports 47, 197 (2002).6. J. Tamayo and R. Garcia, Appl. Phys. Lett. 73, 2926 (1998).7. A. San Paulo and R. Garcia, Phys. Rev. B 64, 193411 (2001).8. T. D. Stowe, T. W. Kenny, D. J. Thomson, and D. Rugar, Appl. Phys.Lett. 75, 2785 (1999).9. P. Grütter, Y. Liu, P. LeBlanc, and U. Dürig, Appl. Phys. Lett. 71,279 (1997).10. Martin Stark, Reinhard Guckenberger, Andreas Stemmer, and Robert W.Stark, J. Appl. Phys. 98, 114904 (2005).11. A. G. Onaran, M. Balantekin, W. Lee, W. L. Hughes, B. A. Buchine, R.O. Guldiken, Z. Parlak, C. F. Quate, and F. L. Degertekin, Rev. Sci.Instrum. 77, 023501 (2006).12. M. Dienwiebel, E. de Kuyper, L. Crama, J. W. M. Frenken, J. A.Heimberg, D.-J. Spaanderman, D. Glastra van Loon, T. Zijistra and E. vander Drift, Rev. Sci. Instr. 76, 043704 (2005).

13. Newnham, Properties of Materials: Anisotropy, Symmetry, Structure,New York, Oxford University Press, (2005). 14. Fukada, Biorheology 32,593 (1995).

15. P. Güthner and K. Dransfeld, Appl. Phys. Lett. 61, 1137 (1992).16. Nanoscale Characterization of Ferroelectric Materials, ed. M. Alexeand A. Gruverman, Springer (2004).17. Nanoscale Phenomena in Ferroelectric Thin Films, ed. Seungbum Hong,Kluwer (2004).18. B. J. Rodriguez, A. Gruverman, A. I. Kingon, R. J. Nemanich, and O.Ambacher, Appl. Phys. Lett. 80, 4166 (2002).19. S. V. Kalinin, B. J. Rodriguez, S. Jesse, T. Thundat, and A.Gruverman, Appl. Phys. Lett. 87, 053901(2005).20. V. Likodimos, M. Labardi, and M. Allegrini, Phys. Rev. B 66, 024104(2002).21. C. Harnagea, M. Alexe, D. Hesse, and A. Pignolet, Appl. Phys. Lett.83, 338 (2003).22. H. Okino, J. Sakamoto, and T. Yamamoto, Jpn. J. Appl. Phys., Part 142, 6209 (2003).23. Rabe, Atomic Force Acoustic Microscopy, in Applied Scanning ProbeMethods, Vol II, Eds. B. Bhushan and H. Fuchs, Springer, N.Y. (2006).

24. S. Jesse, A. P. Baddorf, and S. V. Kalinin, Nanotechnology 17, 1615(2006).

25. J. E. Sader, J. Appl. Phys. 84, 64 (1998).26. R. Garcia and R. Pérez, Surf. Sci. Reports 47, 197 (2002).

27. D. Sarid, Scanning Force Microscopy, Oxford University Press, NewYork (1991).

28 K. Yamanaka, Y. Maruyama, T. Tsuji, and K. Nakamoto, Appl. Phys.Lett. 78, 1939 (2001).

1.-18. (canceled)
 19. An apparatus, comprising: a band excitation signalgenerator that generates an excitation signal; a probe coupled to theband excitation signal generator wherein the probe is simultaneouslyexcited at a plurality of frequencies within a predetermined frequencyband based on the excitation signal; and a detector coupled to the probethat measures a response of the probe across a subset of frequencies ofthe predetermined frequency band.
 20. The apparatus of claim 19, whereinthe detector scans a sample held in the apparatus, the detector measuresthe response of the probe at each position crossed during the scan. 21.The apparatus of claim 20 further comprising a relevant dynamicparameter extractor coupled to the detector, the relevant dynamicparameter extractor includes a processor that performs a mathematicaltransform on the response and outputs an amplitude-frequency data andphase-frequency data at each position of the sample scanned by thedetector.
 22. The apparatus of claim 21, wherein the mathematicalfunction is selected from the group consisting of an integral transformand a discrete transform.
 23. The apparatus of claim 21, wherein therelevant dynamic parameter extractor extracts resonant frequency,maximum amplitude, and Q factor parameters for each position of thesample based on an analysis of the amplitude frequency data and thephase-frequency data. 24.-29. (canceled)
 30. The apparatus of claim 23,wherein the relevant dynamic parameter extractor extracts the resonantfrequency, the maximum amplitude, and the Q factor parametersindependently for each position.
 31. The apparatus of claim 19, furthercomprising an active operational feedback component that adjusts theexcitation signal based on the response measured by the detector. 32.The apparatus of claim 19, wherein the subset of frequencies of thepredetermined frequency band includes a selected frequency andassociated resonance frequencies.
 33. The apparatus of claim 19, whereinthe subset of frequencies is substantially the same as the predeterminedfrequency band.
 34. The apparatus of claim 19, wherein the excitationsignal includes a controlled amplitude and phase density within thepredetermined frequency band.
 35. An apparatus, comprising: a bandexcitation signal generator that generates an excitation signal; a probecoupled to the band excitation signal generator wherein the probe issimultaneously excited at a plurality of frequencies within apredetermined frequency band based on the excitation signal; a detectorcoupled to the probe that measures a response of the probe across asubset of frequencies of the predetermined frequency band; and an activeoperational feedback component that adjusts the excitation signal basedon the response measured by the detector.
 36. The apparatus of claim 35further comprising a relevant dynamic parameter extractor coupled to thedetector and the active operational feedback component, the relevantdynamic parameter extractor includes a processor that performs amathematical transform on the response and outputs anamplitude-frequency data and phase-frequency data at each position ofthe sample scanned by the detector.
 37. The apparatus of claim 36,wherein the mathematical function is selected from the group consistingof an integral transform and a discrete transform.
 38. The apparatus ofclaim 36, wherein the relevant dynamic parameter extractor extractsresonant frequency, maximum amplitude, and Q factor parameters for eachposition of the sample based on an analysis of the amplitude frequencydata and the phase-frequency data.
 39. The apparatus of claim 38,wherein the relevant dynamic parameter extractor extracts the resonantfrequency, the maximum amplitude, and the Q factor parametersindependently for each position.
 40. The apparatus of claim 35, whereinthe subset of frequencies of the predetermined frequency band includes aselected frequency and associated resonance frequencies.
 41. Theapparatus of claim 35, wherein the subset of frequencies issubstantially the same as the predetermined frequency band.
 42. Theapparatus of claim 35, wherein the excitation signal includes acontrolled amplitude and phase density with the predetermined frequencyband.
 43. An apparatus, comprising: a band excitation signal generatorthat generates an excitation signal; a probe coupled to the bandexcitation signal generator wherein the probe is simultaneously excitedat a plurality of frequencies within a predetermined frequency bandbased on the excitation signal; a detector coupled to the probe thatscans a sample held in the apparatus to measure a response of the probeacross a subset of frequencies of the predetermined frequency band; anda relevant dynamic parameter extractor that separately extracts resonantfrequency, maximum amplitude, and Q factor parameters associated witheach position crossed during the scan.
 44. The apparatus of claim 43,wherein the relevant dynamic parameter extractor is coupled to thedetector, the relevant dynamic parameter extractor includes a processorthat performs a mathematical transform on the response of the probe andoutputs an amplitude-frequency data and phase-frequency data at eachposition of the sample scanned by the detector.
 45. The apparatus ofclaim 44, wherein the mathematical function is selected from the groupconsisting of an integral transform and a discrete transform.
 46. Theapparatus of claim 44, wherein the relevant dynamic parameter extractorextracts the resonant frequency, the maximum amplitude, and the Q factorparameters for each position of the sample based on an analysis of theamplitude frequency data and the phase-frequency data.
 47. The apparatusof claim 43, wherein the relevant dynamic parameter extractor extractsthe resonant frequency, the maximum amplitude, and the Q factorparameters independently for each position.
 48. The apparatus of claim43, wherein the subset of frequencies of the predetermined frequencyband includes a selected frequency and associated resonance frequencies.49. The apparatus of claim 43, wherein the subset of frequencies issubstantially the same as the predetermined frequency band.
 50. Theapparatus of claim 43, wherein the excitation signal includes acontrolled amplitude and phase density with the predetermined frequencyband.
 51. The apparatus of claim 43 further comprising an activeoperational feedback component that adjusts the excitation signal basedon the extracted resonant frequency, maximum amplitude, and Q factorparameters